When people enter an apple store,
Imagine we sample the population and try to obtain p from the samples.
Now P becomes uncertain, it is random variable
According to the Central Limit Theorem, for large samples, the sample proportion is approximately normally distributed, with mean:
Where:
If we are investingating a mean(not a proportion) the formula for standard deviation is:
Sometimes we want to compute the probability of successes being more than a certain number.
We know that we can get the area under a curve by using the z-scores, but this distribution only approximates a normal distribution when using a large n.
So when we have a small n we need to go sideways:
For the population of individuals who own an iPhone, suppose p = 0.25 is the proportion that has a given app.
Find the probability that the proportion of having the app is at least 0.75 when n = 4.
Here the sample size is too small, so we can't use the normal distribution stuff.
0.75 of 4 = 3, so we need the probability that at least 3 people have the app.
We do that by summing the probabilities that 3 people have the app and 4 people have the app.
Since those probabilities are discrete and there are only 2 possible outcomes per trial, we can use the binomial distribution formula
In the population, IQ scores are normally distributed with mean µ = 100 and variance σ 2 = 15. Suppose to draw a random samples of 25 individuals from the population and measure the IQ score
The standard deviation of the sample mean for a sample of size 25. Is given by:
The z-scores for 98 and 102 are:
The areas given by the z-tables for the z-scores are:
The area between the two z-scores is given by: